In APL and its relatives (en.wikipedia.org/wiki/APL_(programming_language))every function is right associative. It takes a little time to get accustomed to that, but once you have got it you will wonder why on earth people do it differently most of the time.
So, is there a way to extend these concepts to reals like we can put a real exponent can we also "tetrate" real number of times? How about complex tetration?
Suppose the exponent and the base swapped. (let ? be a function st b?a = a^b). This leaves everything unchanged except exponentiation. But this slight adjustment fits with how we write function notation. We typically write f(x) or the function opperated onto an input. So instead of 2+3 meaning starting at 2 and hopping 3 spaces to the right, the change would imply to hop 2 spaces to the right from 3. Likewise 2?3 would mean to square the input, 3. This ? function is left associative as well.
Like how "a to the power of b" is often written as a^b, "a tetrated to the "b is often written as a^^b. Pentation is a^^^b, hexation is a^^^^b, and so on.
That's one "possible" way but the more standard name for it is called the Knuth's up-arrow notation. The up-arrow is not available on a normal keyboard. ^ may be used as a substitution I guess.@@mathcookie8224
Also, please, please post a tutorial on how to create videos with your style. It’s obviously 3blue1brown inspired, but his Python library was never very easy to use. Do you do that, or something else?
Right associative doesn’t even exist? Not as a mathematical concept anyways completely dependent on the specific representation. That’s just how the notation works? Nothing else to explain! Just like why we write functions in Hebrew order. Defined that way. Some intuition but still just a definition.
Thank goodness it's right-associative, or else I would have to write a crazy number of parentheses for Graham's number.
🤣
A video about tetration that assumes you know what it is! No long intro :)
Now you know my channel's style: if you clicked this video, you must already know what a tetration means. I respect your time.
Just want to say, you’re my new favorite small channel. I look forward to watching you grow to 50k then 100k subs over the next years.
Thank you!
In APL and its relatives (en.wikipedia.org/wiki/APL_(programming_language))every function is right associative. It takes a little time to get accustomed to that, but once you have got it you will wonder why on earth people do it differently most of the time.
So, is there a way to extend these concepts to reals like we can put a real exponent can we also "tetrate" real number of times? How about complex tetration?
Yes there is. May cover this topic in the future as well.
Simple and to the point
Suppose the exponent and the base swapped. (let ? be a function st b?a = a^b). This leaves everything unchanged except exponentiation.
But this slight adjustment fits with how we write function notation. We typically write f(x) or the function opperated onto an input.
So instead of 2+3 meaning starting at 2 and hopping 3 spaces to the right, the change would imply to hop 2 spaces to the right from 3. Likewise 2?3 would mean to square the input, 3. This ? function is left associative as well.
What is the keyboard character notation for tetration?
I think you can only do it using LaTeX for the common notations, unfortunately.
Like how "a to the power of b" is often written as a^b, "a tetrated to the "b is often written as a^^b. Pentation is a^^^b, hexation is a^^^^b, and so on.
That's one "possible" way but the more standard name for it is called the Knuth's up-arrow notation. The up-arrow is not available on a normal keyboard. ^ may be used as a substitution I guess.@@mathcookie8224
What operations you get for negative n? Rational n? Real n? Complex n?
Awesome question. This is an active research area. I haven’t looked into it.
That the universe is too small for writing things out all the way re H5 gave me the biggest chuckle.
Also, please, please post a tutorial on how to create videos with your style. It’s obviously 3blue1brown inspired, but his Python library was never very easy to use. Do you do that, or something else?
Maybe sometime in the future :)
Something something Ackermann's functions?
They have similar mathematical taste.
Sadly the right associativity makes it annoying to extend tetration to the reals
(a^^b)^^(a^^c) != a^^(b^c)
Right associative doesn’t even exist? Not as a mathematical concept anyways completely dependent on the specific representation. That’s just how the notation works? Nothing else to explain! Just like why we write functions in Hebrew order. Defined that way. Some intuition but still just a definition.